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Operator Methods in Quantum Mechanics

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ISBN-10: 0486425479

ISBN-13: 9780486425474

Edition: 2002

Authors: Martin Schechter

List price: $18.95
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Description:

This advanced undergraduate and graduate level text introduces the power of operator theory as a tool in the study of quantum mechanics, assuming only a working knowledge of advanced calculus and no background in physics.
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Book details

List price: $18.95
Copyright year: 2002
Publisher: Dover Publications, Incorporated
Publication date: 2/3/2003
Binding: Hardcover
Pages: 352
Size: 4.75" wide x 8.50" long x 0.75" tall
Weight: 0.748
Language: English

Preface
Acknowledgments
A Message to the Reader
List of Symbols
One-Dimensional Motion
Position
Mathematical Expectation
Momentum
Energy
Observables
Operators
Functions of Observables
Self-Adjoint Operators
Hilbert Space
The Spectral Theorem
Exercises
The Spectrum
The Resolvent
Finding the Spectrum
The Position Operator
The Momentum Operator
The Energy Operator
The Potential
A Class of Functions
The Spectrum of H
Exercises
The Essential Spectrum
An Example
A Calculation
Finding the Eigenvalues
The Domain of H
Back to Hilbert Space
Compact Operators
Relative Compactness
Proof of Theorem 3.7.5
Exercises
The Negative Eigenvalues
The Possibilities
Forms Extensions
The Remaining Proofs
Negative Eigenvalues
Existence of Bound States
Existence of Infinitely Many Bound States
Existence of Only a Finite Number of Bound States
Another Criterion
Exercises
Estimating the Spectrum
Introduction
Some Crucial Lemmas
A Lower Bound for the Spectrum
Lower Bounds for the Essential Spectrum
An Inequality
Bilinear Forms
Intervals Containing the Essential Spectrum
Coincidence of the Essential Spectrum with an Interval
The Harmonic Oscillator
The Morse Potential
Exercises
Scattering Theory
Time Dependence
Scattering States
Properties of the Wave Operators
The Domains of the Wave Operators
Local Singularities
Exercises
Long-Range Potentials
The Coulomb Potential
Some Examples
The Estimates
The Derivatives of V(x)
The Relationship Between X[subscript t] and V(x)
An Identity
The Reduction
Mollifiers
Exercises
Time-Independent Theory
The Resolvent Method
The Theory
A Simple Criterion
The Application
Exercises
Completeness
Definition
The Abstract Theory
Some Identities
Another Form
The Unperturbed Resolvent Operator
The Perturbed Operator
Compact Operators
Analytic Dependence
Projections
An Analytic Function Theorem
The Combined Results
Absolute Continuity
The Intertwining Relations
The Application
Exercises
Strong Completeness
The More Difficult Problem
The Abstract Theory
The Technique
Verification for the Hamiltonian
An Extension
The Principle of Limiting Absorption
Exercises
Oscillating Potentials
A Surprise
The Hamiltonian
The Estimates
A Variation
Examples
Exercises
Eigenfunction Expansions
The Usefulness
The Problem
Operators on L[superscript p]
Weighted L[superscript p]-Spaces
Extended Resolvents
The Formulas
Some Consequences
Summary
Exercises
Restricted Particles
A Particle Between Walls
The Energy Levels
Compact Resolvents
One Opaque Wall
Scattering on a Half-Line
The Spectral Resolution for the Free Particle on a Half-Line
Exercises
Hard-Core Potentials
Local Absorption
The Modified Hamiltonian
The Resolvent Operator for H[subscript 1]
The Wave Operators W[subscript plus or minus] (H[subscript 1] H[subscript 0])
Propagation
Proof of Theorem 14.5.1
Completeness of the Wave Operators W[subscript plus or minus], (H[subscript 1] H[subscript 0])
The Wave Operators W[subscript plus or minus] (H, H[subscript 1])
A Regularity Theorem
A Family of Spaces
Exercises
The Invariance Principle
Introduction
A Simple Result
The Estimates
An Extension
Another Form
Exercises
Trace Class Operators
The Abstract Theorem
Some Consequences
Hilbert--Schmidt Operators
Verification for the Hamiltonian
Exercises
The Fourier Transform
Exercises A
Hilbert Space
Exercises B
Holder's Inequality and Banach Space
Bibliography
Index